Another way to extend the Complex Plane (Insertsomethingthatgetsyourattention)

In summary, the conversation discusses the proposal of a new number J that could potentially extend the complex plane to a third dimension. The equation Logb(J) = -b is suggested as a solution, but it is debated whether any existing algebraic extension, such as Quaternions or Octonions, can satisfy this equation. The potential form of this new number is also discussed, with examples given as a + bi + cj and a + bi + cj + dji. The conversation ends with an invitation for further thoughts and creative responses on the topic.
  • #1
Frogeyedpeas
80
0
Hey guys so I was thinking about how to extend the Complex Plane out to a third dimension and I started reading the whole tidbit about Quaternions and their mechanics when I realized that I want to propose a whole new question. Now please feel free to prove me wrong if you can answer it because I haven't found a whole lot.

Imagine a number J who satisfies the solution to the following equation

Logb(J) = -b

FOR ALL B:

There is no complex number that satisfies that solution and I believe (as uneducated as I might be in this subject) that there is no Quaternion, Octonion or any type of standard Algebraic extension of the number line that satisfies this equation. If this number J can be proposed as the new number extension to the complex plane, then,

We get numbers being described in the form of:

a + bi + cj. Now keeping in mind the ability for numbers to cross:

a + bi + cj + dji is what this number can look like...

What's your take on it?
 
Last edited:
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  • #2
Any replies, opinions, haikus, limericks, epic poetries, hard rock songs, and qwerty-piano recitals would be cool
 
  • #3
Ok... if by Logb you mean log in the base b, then we have:
[tex]log_b(J)=-b[/tex]
[tex]\frac{ln(J)}{ln(b)}=-b[/tex]
[tex]ln(J)=-b ln(b)[/tex]
[tex]J=e^{-b ln(b)}[/tex]
[tex]J=(e^{ln(b)})^{-b}[/tex]
[tex]J=b^{-b}[/tex]
Which is, at worst, a complex number (if b were complex). If b were real, so is J.
 
  • #4
If you want a number J to satisfy that equation, you are not extending the complex plane, you are reducing it by the equvalence relation [tex]b^{-b} = c^{-c}[/tex] for all complex b and c. If the function z^{-z} := e^{-z\log(z)} for some branch of the logarithm is surjective, you will have collapsed the complex plane to a single point.
 
  • #5


I find this proposal intriguing. It seems to suggest the possibility of a new number system that could potentially extend the Complex Plane into a third dimension. However, before jumping to any conclusions, we would need to thoroughly investigate and understand the properties and implications of this proposed number J. We would also need to explore if this number can be applied in a meaningful way in mathematical and scientific contexts. Additionally, we would need to determine if this number can be consistently defined and if it follows the same rules and operations as other numbers in the Complex Plane. Overall, while this proposal is certainly thought-provoking, it would require further research and analysis before being considered a valid extension of the Complex Plane.
 

1. What is the Complex Plane?

The Complex Plane is a mathematical concept that represents complex numbers on a two-dimensional plane. It is composed of an x-axis and a y-axis, with the y-axis representing the imaginary part of a complex number and the x-axis representing the real part.

2. How is the Complex Plane extended?

The Complex Plane can be extended by adding a third dimension, the z-axis, which represents the magnitude or modulus of a complex number. This creates a three-dimensional space known as the Extended Complex Plane.

3. What is the purpose of extending the Complex Plane?

Extending the Complex Plane allows for a more comprehensive representation of complex numbers, including their magnitude. It also allows for a better understanding of complex functions and their behavior in three-dimensional space.

4. What are some applications of the Extended Complex Plane?

The Extended Complex Plane has various applications in mathematics, physics, and engineering. It is used in the study of complex functions, vector calculus, and differential equations. It is also used in fields such as electromagnetism and fluid dynamics.

5. Are there any limitations to the Extended Complex Plane?

One limitation of the Extended Complex Plane is that it can be difficult to visualize and work with in higher dimensions. Additionally, it may not always be necessary to extend the Complex Plane, and the original two-dimensional representation may suffice for certain applications.

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